Affine Hecke algebras and generalized standard Young tableaux

Abstract

This paper introduces calibrated representations for affine Hecke algebras and classifies and constructs all finite dimensional irreducible calibrated representations. The primary technique is to provide indexing sets for controlling the weight space structure of finite dimensional modules for the affine Hecke algebra. Using these indexing sets we show that (1) irreducible calibrated representations are indexed by skew local regions, (2) the dimension of an irreducible calibrated representation is the number of chambers in the local region, (3) each irreducible calibrated representation is constructed explicitly by formulas which describe the action of the generators of the affine Hecke algebra on a specific basis in the representation space. The indexing sets for weight spaces are generalizations of standard Young tableaux and the construction of the irreducible calibrated affine Hecke algebra modules is a generalization of A. Young's seminormal construction of the irreducible representations of the symmetric group. In this sense Young's construction has been generalized to arbitrary Lie type.